Summary: Li-Yau Type Gradient Estimates and Harnack
Inequalities by Stochastic Analysis
Marc Arnaudon and Anton Thalmaier
Abstract.
In this paper we use methods from Stochastic Analysis to establish
Li-Yau type estimates for positive solutions of the heat equation. In
particular, we want to emphasize that Stochastic Analysis provides nat-
ural tools to derive local estimates in the sense that the gradient bound
at given point depends only on universal constants and the geometry
of the Riemannian manifold locally about this point.
§1. Introduction
The effect of curvature on the behaviour of the heat flow on a Rie-
mannian manifold is a classical problem. Ricci curvature manifests itself
most directly in gradient formulas for solutions of heat equation.
Gradient estimates for positive solutions of the heat equation serve
as infinitesimal versions of Harnack inequalities: by integrating along
curves on the manifold local gradient estimates may be turned into local
Harnack type inequalities.
Solutions to the heat equation
(1.1)